Linear Functions Flashcards

A linear function f satisfies f(3) = 7 and f(7) = 3. What is the value of f(f(5))?

Answer: 5

The slope is (3 − 7)/(7 − 3) = −1. Using point (3, 7): f(x) = −x + 10. So f(5) = 5, and f(f(5)) = f(5) = 5. The value 5 is a fixed point of the function.

The graph of line ℓ is perpendicular to the graph of 4x − 6y = 12 and passes through the point (2, −1). Which of the following is an equation of line ℓ?

Answer: y = −(3/2)x + 2

Rewrite 4x − 6y = 12 as y = (2/3)x − 2, giving slope 2/3. A perpendicular slope is −3/2. Through (2, −1): y + 1 = −(3/2)(x − 2) → y = −(3/2)x + 2.

The system ax + 2y = 6 and 3x + by = 9 has infinitely many solutions. Which condition must be true?

Answer: a = 2 and b = 3

For infinitely many solutions, the equations must be scalar multiples: a/3 = 2/b = 6/9 = 2/3. So a = 2 and b = 3. Check: 2x + 2y = 6 is (2/3) times 3x + 3y = 9 ✓.

A table of values for a linear function g is shown: x = −5, −2, 1, 4 and g(x) = 14, 8, 2, −4. What is the value of g(0)?

Answer: 4

The slope is (8 − 14)/(−2 − (−5)) = −6/3 = −2. Using point (1, 2): g(x) = −2x + 4. Therefore g(0) = 4.

A rental company charges a flat hourly rate and a per-mile rate. A customer who drives 12 miles in 3 hours pays $24. Another who drives 20 miles in 2 hours pays $28. What is the per-mile charge?

Answer: $1.00

Let a = hourly rate, b = per-mile rate. System: 3a + 12b = 24 and 2a + 20b = 28. Multiply first by 2 and second by 3: 6a + 24b = 48 and 6a + 60b = 84. Subtracting: 36b = 36, so b = $1.00 per mile.

Line m passes through (−3, k) and (5, 2k − 1). If the slope of line m is 3/4, what is the value of k?

Answer: 7

Slope = (2k − 1 − k)/(5 − (−3)) = (k − 1)/8 = 3/4. So k − 1 = 6, giving k = 7.